Elucidating the Role of Microstructure in Thiophosphate Electrolytes – a Combined Experimental and Theoretical Study of β‐Li3PS4

Abstract Solid‐state batteries (SSBs) are promising candidates to significantly exceed the energy densities of today's state‐of‐the‐art technology, lithium‐ion batteries (LIBs). To enable this advancement, optimizing the solid electrolyte (SE) is the key. β‐Li3PS4 (β‐LPS) is the most studied member of the Li2S‐P2S5 family, offering promising properties for implementation in electric vehicles. In this work, the microstructure of this SE and how it influences the electrochemical performance are systematically investigated. To figure this out, four batches of β‐LPS electrolyte with different particle size, shape, and porosity are investigated in detail. It is found that differences in pellet porosities mostly originate from single‐particle intrinsic features and less from interparticle voids. Surprisingly, the β‐LPS electrolyte pellets with the highest porosity and larger particle size not only show the highest ionic conductivity (up to 0.049 mS cm–1 at RT), but also the most stable cycling performance in symmetrical Li cells. This behavior is traced back to the grain boundary resistance. Larger SE particles seem to be more attractive, as their grain boundary contribution is lower than that of denser pellets prepared using smaller β‐LPS particles.


Supplementary Information (SI)
Particle Characterization

Lattice Parameters
The calculated lattice parameters from XRD and the theoretical density of the different LPS batches are shown in   (7) 6.1344 (5) 640.69 (12) 1.866

XPS
To analyse the chemical surface composition of the different batches X-ray photoelectron spectroscopy (XPS) was used. The measurements were performed using a monochromatic Al Kα source (hν = 1486.6 eV) and a Phoibos 150 XPS spectrometer with a microchannel plate and a delay line detector (DLD) in fixed analyser transmission mode. High-resolution scans were collected at 400 W, 30 eV pass energy, and 0.1 eV energy step. The photoelectron spectra were calibrated using the hydrocarbons peak (-C-C-/-C-H-) at 284.8 eV as reference.
The data evaluation was carried out with CASAXPS software using a nonlinear Shirley-type background and Voight profile function.
The spectra of the four -LPS batches are shown in For clarity, the Li 1s, P 2p, S 2p, C 1s and O 1s spectra are plotted separately (from left to right). In all samples the S 2p spectra features the S 2doublet peaks (2p 1/2 and 2p 2/3 ) of Li 3 PS 4 at 161.5 eV and 162.5 eV, respectively. [2,3,4] Additionally a peak toward higher binding energies appears, which can be assigned to adsorbed H 2 S [5] and appears particularly pronounced for B2 and B3. In fact, B2 and B3 show 2-3 times higher H 2 S content (3.1% and 3.9%) compared to B1 and B4 (1.1% and 1.4%). On the other hand, B1 and B4 have accordingly higher content of pure Li 3 PS 4 at the surface (27.3% and 25.8%). The P 2p spectra are in accordance with the S 2p spectra, confirming the doublets appearing at 133.3 eV and 132.5 eV assigned to PS 4 3-. [2,3,4] Besides the expected P and S compounds also C and O compounds could be detected on the electrolyte surface of all four batches. The surface of the SE contains products like e.g., ethers (-COC-), alcohols(-COH), and carboxyl/esters (-COOH/-COOR) originating, most probably, from reaction with the atmosphere. [6] But having a closer look at the O 1s and Li 1s spectra of B2 and B3, lithium hydroxide, LiOH, could be detected, whereas the surface of B3 contains the highest amount on LiOH (4.20%). [7] As we know, that LPS is sensitive towards humidity [8] , we can assume that B2 and B3 have been in contact with moisture triggering the formation of LiOH compounds on the surface, also in accordance with the increased content of H 2 S for these SE batches. [9] Surface impurities play a crucial role on the interface formation with Li metal, though, later, we will see that the influence of the different surface impurities of the SE batches seem to have a not significant influence on the electrochemical performance.

SEM -Particle Sizes
In Fig.SI.2, higher magnification SEM images of all 4 batches are shown. The particles size was determined by Image J. The secondary particles were considered for the given average particle sizes of 20.69 ± 3.81 µm for B1 (20 µm), 6.90 ± 1.67 µm for B2 (7 µm) and 3.71 ± 0.89 µm for B3 (3µm) and 5.53 ± 1.14 µm B4 (3 µm). B3 and B4 do not display the characteristic rod-like particle shape, the secondary particle sizes are therefore much smaller, and they tend to easily form agglomerates.

SEM
The cross-sectional images of a B1 pellet pressed under active vacuum are shown in Fig.SI.3 a-b). The comparison with Fig. 4a shows that no further densification was achieved.
A single LPS particle from B1 was then analysed further. The cross-sectional images shown in Fig SI.3 c-d) reveals the existence of a large porosity between single crystallites within the single particle. Such intrinsic porosity is, apparently, not removed by cold-pressing. magnification. SEM images of c) pristine B1 particle, d) cross section of a B1 particle at low magnification and e) at high magnification.

SAXS
The SAXS patterns were collected using a Xeuss 3.0c (Xenocs -Grenoble, France) equipped with an Eiger2 1M detector. The Sample-to-Detector distance was set to 1100 mm. A Mo Kα source was exploited with a beam size of 0.35x0.35 mm 2 , obtaining a flux of ~10 5 photons per sec. The pellet samples where sticked on a perforated metal plate, but the optic path of the X-rays was not covered with tape. The sample chamber of the instrument was kept under vacuum (P=15 μbar) during the experiment. Each pellet sample was exposed for 20 minutes to X-rays to ensure a good signal-to-noise ratio. The tape background was also collected and subtracted from the total scattering curve. Data treatment was performed with the SAXSutilities2 software.
In obtain two different slopes, resulting from two different pore shapes, oblates and disks.

Microstructure Generation
The SEM analysis indicates that the particle morphology and size for the four LPS batches varies significantly. Notably, for B1 the estimated average particle diameter of 20 µm reduces to 7 µm for B2 and decreases for B3 and B4 further down to 4-5 µm. Another indication for the reducing particle size is the increased surface area measured via BET. The measurements show an increase by a factor of two between B1-B2 and B3-B4. The combination of both experimental results is taken as a starting point to investigate the influence of particle morphology on the pellet resistance. As shown in Figure SI.3, the particle surface is assumed to be sufficiently ductile and robust to allow a dense palletisation without reducing the internal particle fibrous structure. Three different SE microstructures were generated with varying particle diameter from 20 µm, 7 µm to 3 µm in agreement with the experiment. The microstructure generation is performed via GeoDict [10] and the respective grain segmentation step is later processed in Matlab. Starting with the GeoDict structure generator, we approximate the SE particles

Particle Size Distribution
The particle size distribution is estimated during the segmentation step, at which the number of created particles and their respective voxel volume ( ) is listed. The particle diameter, , of a spherical particle is approximated with the same volume, which results in the following relation: By applying Eq-SI.1, we can calculate the respective particle diameter. µm, b) 7 µm, and c) 3 µm. Note that the Gaussian distribution fit for the mean value estimation is shown in solid red. As expected, the number of particles reduces with increasing particle diameter. Further structural parameters of the samples are given in Table SI.1. Note, that each LPS batch is assigned to a specific microstructure based on the SEM analysis. In particular, the 20 µm particle structure corresponds to B1, the 7 µm particle structure to B2 and the 3 µm particle structure to B3 and B4. Since the SEM images have a limited resolution and are only 2D, this assignment is not unique. Therefore, the obtained particle size distributions could potentially underestimate particle inhomogeneities or overestimate local effects.

Effective Bulk Conductivity Variation
The change in the effective bulk conductivity , of the porous particles is calculated based on the variation in porosity ( ) and tortuosity ( ) by using the Bruggeman relation , = . ( The porosity is varied from 10% to 40% and the tortuosity from 1.3 to 2.0 to reproduce the experimentally measured particle properties. The resulting effective bulk conductivities calculated for the microstructure simulation study are given in

Microstructure Transport Modell BEST
A prominent advantage of the microstructure resolved simulations is the ability to incorporate inherently the morphological properties of materials and components in the battery cell, such as the particle size and electrode tortuosity. The battery simulation studies are performed on virtually generated voxel-based electrodes containing standard experimental microstructural parameters like pellet thickness and particle size. In Table SI.3, the relevant transport equations implemented in the framework BEST [12] are summarised: Details on the modelling equations, the respective derivation, and further applications of the presented all-solid-state battery model are given in Ref. [11,13] . Specific performance and interface studies on thiophosphate-based composite cathodes can be found in Ref [14,15] . batches. The resolved pellets cross-sections suggest that the particle morphology remains present in the compressed pellets. A schematic representation of the particle distribution is given in Fig.1b) of the main study, where each colour represents a single particle in the SE structure. This assumption justifies the application of a modified GB interface flux model to account for the additional GB resistance in the compressed SE.

Grain Boundary Interface Flux
Due to the high numerical cost of spatially resolving the nanometer scale GB phase at the grain-grain contact area inside the microstructure, the additional transport resistance is accounted by using a modified GB interface flux expression: The grain-grain interface current is defined by the lithium hopping rate and the respective grain concentrations of lithium ̃ . The driving force for the lithium hopping is given through the electrochemical potential difference between the two individual grains.
The resulting GB interface flux is used to describe the lithium hopping process at the grain boundary. The interface flux expression is defined at each grain interface of the segmented virtual SE structure. In Fig.1a), we visualised the extend of the grain-grain interface area inside the segmented SE pellet.
Through this approach, we can account for resistive GB contributions at the grain-grain transition and enable microstructure resolved cell simulations at reasonable time scales. Note that the parameter is adjusted through experimental EIS measurements and therefore approximates the transport energy barrier for the lithium and additional morphological effects of the grain-grain contact. A detailed study of the presented model for an oxide-based porous SE network is given in Ref. [11] In the following, we further summarise the GB hopping rate ℎ and constant concentration contribution through the GB exchange current density = ℎ .
The parametrisation of the GB interface flux model is based on the impedance measurement of LPS B1, since the high conductivity and the particle structure indicates the lowest influence of the GB resistance. We keep the GB interface parametrisation fixed for the three studied SE microstructures.

Model Parameterization
The following Table SI.4 summarises the transport parameters and thermodynamic parameters of the simulated battery cell. The estimation of the transport parameters of -LPS is based on the deconvolution of corresponding electrochemical impedance measurements of symmetric cells under blocking conditions. A detailed discussion can be found in Ref. [11] . The general material parameters such as diffusion and conductivity were either measured or taken from literature. Half Cell Analysis

EIS EIS: Blocking Configuration
To determine the ionic conductivity of the four β-LPS batches and to analyse their interface with the lithium metal electrode, electrochemical impedance analysis was conducted. To determine the ionic conductivity, for each temperature, the R g+gb was calculated by fitting the EIS curves with an equivalent circuit consisting of (R (1) )-(R (2) C (2) )-(CPE), whereas R 1 was considered as R grain (grain) and R 2 is considered as R gb (grain boundary). Further, C is the capacitance and CPE the constant phase element. Some representative EIS curves @20°C are shown in Fig.SI.8).
As both, R grain and R gb have an influence on the ionic conductivity, the sum of both values is used to calculate the ionic conductivities. By inserting the resulting R g+gb values into the equation-SI.4, the ionic conductivities at each temperature are calculated. Where ionic conductivity, d = thickness of the SE pellet, S = area of electrodes contacting SE pellet, and = grain+ grain boundary resistance. The obtained resistances for grain and GB are given in Table SI.5. The resistance mean value and standard deviation calculation is based on four separate cells/ measurements. The relative ratio between and is taken as an indicator for the dominating transport process. We observe that the ratio constantly decreases from 0.19 to 0.10 down to 0.08 for LPS B3 and B4. The ratio is a measure for the relative increase in GB resistance as the particle size decreases. This trend is in line with our assumption that the conductivity reduction is mainly dominated by the increase in GB resistance contribution with reduced LPS particle size. Furthermore, we estimate the interface resistance per GB based on a geometric approximation of the LPS sample with varying particle sizes. The single interface GB resistance is approximated as: Using a similar approach, we calculate our microstructure model's respective GB interface resistance. The simulated GB resistance Note that we selected the effective conductivity values from Whereas σ is the ionic conductivity, T is temperature, A is a pre-exponential factor, n is the concentration of mobile-ion carriers, E a is the activation energy of thermally activated process and k b is the Boltzmann constant.
The calculated values for the ionic conductivities @20°C and the values for the activation energies are presented in Table SI.7.   The applied equivalent circuit in this case consists of (R (1) )-( R (2) C (2) )-(R (3) C (3) )-W. Whereas R 1 is representative as R g , R 2 is considered as R gb and R 3 is standing for R int (interfacial resistance), which is determined by R CT (Charge Transfer) and R SEI (Solid Electrolyte interface). Whereas W is standing for Warburg element. During the aging experiment shown in Fig.7, the interface resistance R int is evolving overtime due to reactivity between LPS and the Li metal. The R and C values for the day 1 of aging are reported in Table.SI.8.   The impedance simulation setup is adapted from the experimental lab system using steel stamps to create blocking electrode conditions. The interface parametrisation of the virtual blocking steel electrode is adjusted to reproduce the experimentally observed low frequency impedance tail. The respective parameters for the impedance parameterisation in this study are listed in Table SI.4. Details on the impedance simulations and the interface model definition are given in Ref. [18,11] Figure SI.11: Nyquist plot representation of the experimental and simulated impedance response for the three different SE microstructures: a) LPS batch 1 (32% porosity) and 20 µm particle microstructure, b) LPS batch 2 (42% porosity) and 7 µm particle microstructure and c) LPS batch 3 and 4 (5% and 10% porosity) and 3 µm particle microstructure. The dotted lines are showing the experimental values and the simulated impedances are displayed in solid lines. The red colour map gives the variation in tortuosity from 1.3 to 2.0 and the simulated porosity is given by the respective inset. Fig.SI.11 shows the Nyquist plot for the measured impedance of the four LPS batches with steel stamps at 20°C (dotted lines). Moreover, we added the simulated impedance responses at varying particle tortuosity for a fixed particle porosity (straight lines). The virtual porosity value is set to reproduce the experimental pellet porosity (cf. Table 1). The variation of the tortuosity for fixed porosity visualises the qualitative influence on the impedance (colour code). Nevertheless, the agreement between the theoretical and experimental impedance response and the respective conductivity values indicates that the suggested virtual microstructures provide a qualitative representation of the pellet structure.
The simulated impedance response accounts for the GB resistance contribution using the interface flux model. As discussed in the main article, we see that the GB contribution increases with the specific particle surface area for the different structures (cf . Table SI.9).
The high-frequency bulk contribution is currently not described in our model, but the respective resistance contribution and the resulting effective bulk conductivity is given through the constant shift along the real x-axis. Since the resistance contribution depends on the effective bulk conductivity, the shift along the x-axis is changing significantly under variation of the tortuosity and porosity. The GB contribution lies in the high to mid-frequency regime and is given by the fully resolved semi-circle. Additionally, the increasing particle tortuosity also leads to an observable rise in the GB polarisation. In the low frequency regime, the blocking tail behaviour is reproduced. The respective regimes are indicated in It should be noted that in the experimental setup we assume that contributions of the bulk polarization are overlapping with the GB contributions. Therefore the deconvolution of both processes is not straightforward. Thus, the microstructure GB model allows us to selectively model the GB polarization and compare this to the measurements. This provides us with qualitative insight on the relation between the GB resistance, the particle morphology and the GB transport kinetics. For a more detailed description of the utilized EIS modelling approach on solid electrolytes and the respective frequency-dependent effects on the GB transport, we refer to Ref. [11].
We see that the simulated impedance response for B1 and B2 shown in Fig.3a and Fig.3b reproduces the experimental impedance data. Nevertheless, we observe a mismatch in the bulk contribution for B3 and B4 between simulation and experiment. We assume that the selected ionic bulk conductivity in our simulation overestimates the experimental value of B3 and B4, although compared to B1 and B2, the effective conductivity value is higher due to the reduced porosity and tortuosity (see Table SI.2). The mismatch might indicate an additional change in the bulk conductivity for B3 and B4 with decreasing particle size.
Since we use a fixed bulk conductivity value ( = 2.2 ⋅ 10 S cm -1 ) modified by the Bruggeman relation (Equation 2), we do not account for potential bulk conductivity changes which might result from degradation during the assumed milling step of the material or are due to a smaller crystallite size. As shown via SEM analysis, the internal particle porosity is reduced in B3 and B4, which potentially reduces conduction pathways along the surfaces of the material. This interpretation is in line with our observation that the larger particles in B1/B2 have smoother particle surfaces and show less particle agglomeration. The observed particle morphology trend would suggest that the transport along and inside the larger and smoother particles is improved compared to the batches with smaller particles showing stronger surface roughness and agglomeration [19] .
We performed an additional parameter study to investigate the influence of bulk conductivity on the impedance response of B3/B4. Therefore, we selected the 3µm pellet structure and varied the bulk conductivity value from 2.2 ⋅ 10 S cm -1 , 1.5 ⋅ 10 S cm -1 to 1. 0 ⋅ 10 S cm -1 with a fixed tortuosity ( = 1.3) and porosity ( = 0.9). The selected bulk values are based on the experimentally measured LPS conductivities found in the literature [16,17,19] . The  The change in bulk conductivity leads to the assumed shift in the bulk contribution. Again, the decreasing conductivity gives rise to an increase in GB contribution, which results from the fixed GB parameter set we used for all simulation studies. Nevertheless, the agreement between simulation and experiment for B3 and B4 is improved through the reduced bulk conductivity value. We conclude that additional mechanisms might cause the decrease in bulk conductivity between batches B1/B2 and B3/B4. This effect would be in line with the experimental findings. Although batches B1/B2 possesses a higher porosity than B3/B4, the decreased particle size and higher GB contribution and the reduced bulk conductivity lead to decreased effective pellet conductivity. Further experimental and theoretical investigations are necessary to verify our assumption of the decreasing bulk conductivity with decreasing particle size, which is beyond the scope of this study.
We conclude that the presented impedance study provides an additional analysis method to investigate the structural aspects of the SE, such as porosity and particle size, and crosslink them to electrochemical transport phenomena at GB and limit interface phenomena.
Furthermore, since the cell performance is significantly impacted by surface composition and particle morphology, future cell studies will further investigate the aspects.

Frequency-Dependent Impedance Representation
Fig.SI.13 shows the Bode plot representation for the experimental impedance response for LPS batch 1 and the respective simulation with a fixed particle porosity and variation of the internal particle tortuosity. The comparison shows that the high-to mid-frequency regime (10 6 -10 4 Hz) is captured qualitatively through the GB model. In the simulation the high- frequency polarization is partly represented through the constant shift along the y-axis accounting for the effective bulk resistance resulting from tortuosity, porosity and general sample constriction. The observed ideal blocking-tail behaviour in the simulation is shifted towards lower-frequencies due to the assumed perfect contact between SE and current collector. [20] Note that we do not include any specific contact surface modification and therefore can only provide a qualitative description of the capacitive behaviour at low frequencies in the simulations. .

Specific Particle Surface Area
Table SI.9: Calculated structural parameters for the virtual microstructures: average particle diameter and particle specific surface area. The respective particle distributions are given in the supporting information.